scholarly journals Random Homogenization in a Domain with Light Concentrated Masses

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 788
Author(s):  
Gregory A. Chechkin ◽  
Tatiana P. Chechkina
Keyword(s):  
Small Parameter ◽  
Elliptic Problem ◽  
Stochastic Perturbation ◽  
Initial Problem ◽  
Compactness Theorem ◽  
Concentrated Masses ◽  
Homogenization Methods

In the paper, we consider an elliptic problem in a domain with singular stochastic perturbation of the density located near the boundary, depending on a small parameter. Using the boundary homogenization methods, we prove the compactness theorem and study the behavior of eigenelements to the initial problem as the small parameter tends to zero.

scholarly journals On the Question of Asymptotic Integration of Singularly Perturbed Fractional-Order Problems

2018 ◽  
Vol 6 (3) ◽  
Author(s):  
Burkhan Kalimbetov
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Fractional Order ◽  
Initial Problem ◽  
Singularly Perturbed ◽  
Asymptotic Integration ◽  
Partial Derivatives ◽  
Systems Of Differential Equations ◽  
Regularization Problem ◽  
Iterative Systems

In this paper we consider an initial problem for systems of differential equations of fractional order with a small parameter for the derivative. Regularization problem is produced, and algorithm for normal and unique solubility of general iterative systems of differential equations with partial derivatives is given. 

VIBRATIONS OF A MEMBRANE WITH MANY CONCENTRATED MASSES NEAR THE BOUNDARY

1995 ◽  
Vol 05 (05) ◽  
pp. 565-585 ◽  
Author(s):  
MIGUEL LOBO ◽  
EUGENIA PÉREZ
Keyword(s):  
Perturbation Theory ◽  
Asymptotic Behavior ◽  
Small Parameter ◽  
Local Problem ◽  
Micro Structure ◽  
Spectral Perturbation Theory ◽  
Concentrated Masses ◽  
The Masses ◽  
Spectral Perturbation ◽  
Small Eigenvalues

We consider the asymptotic behavior of the vibrations of a membrane occupying a domain Ω ⊂ ℝ2. The density, which depends on a small parameter ε, is of order O(1) out of certain regions where it is O(ε−m) with m>0. These regions, the concentrated masses with diameter O(ε), are located near the boundary, at mutual distances O(η), with η=η(ε)→0. We impose Dirichlet (respectively Neumann) conditions at the points of ∂Ω in contact with (respectively, out of) the masses. Depending on the value of the parameter m(m>2, m=2 or m<2) we describe the asymptotic behavior of the eigenvalues. Small eigenvalues, of order O(εm−2) for m>2, are approached via those of a local problem obtained from the micro-structure of the problem, while the eigenvalues of order O(1) are approached through those of a homogenized problem, which depend on the relation between ε and η. Techniques of boundary homogenization and spectral perturbation theory are used to study this problem.

ON VIBRATIONS OF A BODY WITH MANY CONCENTRATED MASSES NEAR THE BOUNDARY

1993 ◽  
Vol 03 (02) ◽  
pp. 249-273 ◽  
Author(s):  
MIGUEL LOBO ◽  
EUGENIA PEREZ
Keyword(s):  
Asymptotic Behavior ◽  
Small Parameter ◽  
Critical Size ◽  
Local Problem ◽  
Boundary Values ◽  
Micro Structure ◽  
Spectral Convergence ◽  
Fourier And Laplace Transforms ◽  
Concentrated Masses ◽  
The Masses

We consider the asymptotic behavior of the vibration of a body occupying a region Ω⊂ℝ3. The density, which depends on a small parameter ε, is of order O(1) out of certain regions where it is O(ε–m) with m>2. These regions, the concentrated masses with diameter O(ε), are located near the boundary, at mutual distances O(η), with η=η(ε)→0. We impose Dirichlet (respectively Neumann) conditions at the points of ∂Ω in contact with (respectively, out of) the masses. For the critical size ε=O(η2), the asymptotic behavior of the eigenvalues of order O(εm−2) is described via a Steklov problem, where the ‘mass’ is localized on the boundary, or through the eigenvalues of a local problem obtained from the micro-structure of the problem. We use the techniques of the formal asymptotic analysis in homogenization to determine both problems. We also use techniques of convergence in homogenization, Semigroups theory, Fourier and Laplace transforms and boundary values of analytic functions to prove spectral convergence. In the same framework we study the case m=2 as well as the case when other boundary conditions are imposed on ∂Ω.

scholarly journals About Averaging in Hyperbolic Equation under the Influence of Multifrequence Disturbances

2019 ◽  
pp. 9-10
Author(s):  
Yaroslav Bihun ◽  
Ihor Skutar
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Hyperbolic Equation ◽  
Ordinary Differential Equations ◽  
Averaging Method ◽  
Initial Problem

The research deals with the existence of thesolution of the initial problem for hyperbolic equation under themultifrequency disturbances, which are described by the systemof ordinary differential equations (ODE) with multipoint andintegral conditions. The averaging method over fast variables isgrounded and estimation of accuracy of the method whichobviously depends on the small parameter was found.

LAYERED SOLUTIONS WITH CONCENTRATION ON LINES IN THREE-DIMENSIONAL DOMAINS

2014 ◽  
Vol 12 (02) ◽  
pp. 161-194 ◽  
Author(s):  
WEIWEI AO ◽  
JUN YANG
Keyword(s):  
Small Parameter ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Smooth Boundary ◽  
Three Dimensional ◽  
Singularly Perturbed ◽  
Existence Of A Solution ◽  
Straight Line ◽  
Dimensional Domain ◽  
Exponentially Small

We consider the following singularly perturbed elliptic problem [Formula: see text] where Ω is a bounded domain in ℝ3with smooth boundary, ε is a small parameter, 1 < p < ∞, ν is the outward normal of ∂Ω. We employ techniques already developed in [39] to extend their result to three-dimensional domain. More precisely, let Γ be a straight line intersecting orthogonally with ∂Ω at exactly two points and satisfying a non-degenerate condition. We establish the existence of a solution uεconcentrating along a curve [Formula: see text] near Γ, exponentially small in ε at any positive distance from the curve, provided ε is small and away from certain critical numbers. The concentrating curve [Formula: see text] will collapse to Γ as ε → 0.

A Perturbation Theory for the Population of Atomic Energy Levels in Non-Lte Plasmas

1988 ◽  
Vol 102 ◽  
pp. 343-347
Author(s):  
M. Klapisch
Keyword(s):  
Perturbation Theory ◽  
Small Parameter ◽  
Classification Scheme ◽  
Sum Rules ◽  
Energy Levels ◽  
Natural Classification ◽  
Quantum Numbers ◽  
Formal Expansion ◽  
Atomic Energy Levels ◽  
As Effects

AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

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